Life Tests under Competing Causes of Failure and the Theory of Competing Risks

TLDR

In life‑testing, failures are classified into k mutually exclusive causes, each associated with a characteristic life distribution. A general likelihood function accommodating dependent causes and censored data is developed, and maximum‑likelihood estimation is applied to independent Weibull models with equal or unequal shape constants. Applying this method to 121 breast‑cancer patients treated with surgery and/or X‑ray therapy, the estimated Weibull parameters yield the competing‑risk probabilities.

Abstract

Suppose that in a life-testing situation the failure of an individual can be classified into one of k( >1) mutually exclusive classes, usually causes of failure. It is assumed that associated with each cause of failure there is a characteristic life distribution belonging to a specific class of distributions. A general likelihood function is obtained which allows for dependence of the causes, for both censored and uncensored data. The method of maximum likelihood is used to estimate the parameters when the underlying life distributions are independent Weibulls with equal shape constants (the exponential being a special case) and independent Weibulls with unequal shape constants. A study involving 121 patients treated for cancer of the breast by surgery and/or X-ray therapy is analyzed, and from the estimates of the parameters of the underlying life distributions (assumed to be Weibull with unequal shape constants) the relevant probabilities in competing risk theory are obtained.